17.2 Fitted Values and Residuals

• A fitted (or predicted) value is calculated as

$$\hat{y}_{i}=\hat{\beta}_{0}+\hat{\beta}_{1}x_{i,1}+\hat{\beta}_{2}x_{i,2}+\ldots+\hat{\beta}_{k}x_{i,k}$$ - A residual (error) term is calculated as $$e_{i}=y_{i}-\hat{y}_{i},$$ the difference between an actual and a predicted value of $y$. A plot of residuals versus fitted values ideally should resemble a horizontal random band. Departures from this form indicates difficulties with the model and/or data. - Other residual analyses can be done exactly as we did in simple regression. For instance, we might wish to examine a normal probability plot (NPP) of the residuals. Additional plots to consider are plots of residuals versus each $x$ variable separately. This might help us identify sources of curvature or nonconstant variance.